Solve (31/sqrt{3}-1)^2+(1/sqrt{3})^2-1

$$( 3 \frac { 1 } { \sqrt { 3 } } - 1 ) ^ { 2 } + ( \frac { 1 } { \sqrt { 3 } } ) ^ { 2 } - 1$$

Evaluate

$$\frac{10}{3}-2\sqrt{3}\approx -0.130768282$$

Solution Steps

Rationalize the denominator of \frac{1}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\left(3\times \frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}-1\right)^{2}+\left(\frac{1}{\sqrt{3}}\right)^{2}-1$$

The square of \sqrt{3} is 3.

$$\left(3\times \frac{\sqrt{3}}{3}-1\right)^{2}+\left(\frac{1}{\sqrt{3}}\right)^{2}-1$$

Cancel out 3 and 3.

$$\left(\sqrt{3}-1\right)^{2}+\left(\frac{1}{\sqrt{3}}\right)^{2}-1$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{3}-1\right)^{2}.

$$\left(\sqrt{3}\right)^{2}-2\sqrt{3}+1+\left(\frac{1}{\sqrt{3}}\right)^{2}-1$$

The square of \sqrt{3} is 3.

$$3-2\sqrt{3}+1+\left(\frac{1}{\sqrt{3}}\right)^{2}-1$$

Add 3 and 1 to get 4.

$$4-2\sqrt{3}+\left(\frac{1}{\sqrt{3}}\right)^{2}-1$$

Rationalize the denominator of \frac{1}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$4-2\sqrt{3}+\left(\frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\right)^{2}-1$$

The square of \sqrt{3} is 3.

$$4-2\sqrt{3}+\left(\frac{\sqrt{3}}{3}\right)^{2}-1$$

To raise \frac{\sqrt{3}}{3} to a power, raise both numerator and denominator to the power and then divide.

$$4-2\sqrt{3}+\frac{\left(\sqrt{3}\right)^{2}}{3^{2}}-1$$

Subtract 1 from 4 to get 3.

$$3-2\sqrt{3}+\frac{\left(\sqrt{3}\right)^{2}}{3^{2}}$$

The square of \sqrt{3} is 3.

$$3-2\sqrt{3}+\frac{3}{3^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$3-2\sqrt{3}+\frac{3}{9}$$

Reduce the fraction \frac{3}{9} to lowest terms by extracting and canceling out 3.

$$3-2\sqrt{3}+\frac{1}{3}$$

Add 3 and \frac{1}{3} to get \frac{10}{3}.

$$\frac{10}{3}-2\sqrt{3}$$